A051025 Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

26861, 616841, 616849, 616877, 617011, 617269, 617327, 617339, 617359, 617369, 617401, 617429, 617453, 617521, 617537, 617689, 617699, 617717, 622813, 622987, 623003, 623107, 623209, 623299, 623321, 623341, 623353, 623401, 623423, 623437

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

This is a companion sequence to A051024.

Starting from a(27556)=9103362505801 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 418933 terms (see a-file) with a(418933)=9543313015309 as its last term. - Sergei D. Shchebetov, Oct 06 2017

We also discovered the 9th sign-changing zone, which starts from 64083080712569, ends with 64084318523021, and has 13370 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017

We also discovered the 10th sign-changing zone, which starts from 715725135905981, ends with 732156384107921, and has 481194 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 28 2018

Links

• Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
• A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55.
• C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
• C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.54-76.
• M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
• A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
• M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
• Sergei D. Shchebetov, First 418933 terms (zipped file)
• Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Crossrefs

Cf. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547.

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). - Daniel Forgues, Mar 26 2009

Programs

Mathematica

For[i=2; d=0, True, i++, d+=Mod[p=Prime[i], 4]-2; If[d==-1, Print[p]]]
(* Second program: *)
Prime@ Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* Michael De Vlieger, Dec 30 2017 *)

Python

from sympy import nextprime; a, p = 0, 2
while p < 623803:
    p=nextprime(p); a += p%4-2
    if a == -1: print(p, end = ', ')  # Ya-Ping Lu, Jan 18 2025

Extensions

Edited by Dean Hickerson, Mar 10 2002